Pub-lic Good

A bit over three years ago, you were elected president of
the glorious and picturesque country of Molvanîa, a land
untouched by modern dentistry. To secure your landslide victory
in the election you had to make a few promises, some of which,
with the clarity of hindsight, may have been a tad
exaggerated.

Molvanîa’s economy is quite simple compared to that of most
other countries. The two main professions in Molvanîa are
pub-owners, and beer-drinkers. These two groups combined
account for over 75% of the Molvanîan GDP (Gross Domestic
Product). Slightly simplified, the system works like this: the
beer-drinkers borrow money to pay for their beer. This creates
income for the pub-owners. The pub-owners use their income to
purchase AAA-rated bonds, backed by loans to beer-drinkers.
This system is locally referred to as pub-prime lending.

Task

One of your election-time promises was to further optimize
Molvanîa’s Tiger economy through improved city planning. You
have identified a number of suitable construction sites in
which either a pub or a house of a beer-drinker can be built.
There are walkways between some of these sites. To fully
optimize the economy, you want to place buildings such that
each house has at least one pub at only a walkway’s distance,
and each pub has at least one house at only a walkway’s
distance. It might happen that this is impossible, but you will
try your best.

Beware that the city has a quite peculiar lay-out, and it
may not even be possible to draw it on a normal map. Molvanîa
is special that way.

Input

There are $n$
construction sites and $m$
walkways in the city ($1\le n\le
10\, 000$ and $0\le m\le
100\, 000$). The first line contains $n$ and $m$, separated by a single space. The
next $m$ lines contain
integers $x$ and
$y$$(1 \le x, y \le n)$ indicating that
there is a walkway between $x$ and $y$. There are no loops (i.e.,
$x \neq y$) and all lines
with walkway descriptions are distinct.

Output

If it is impossible to build pubs and houses such that every
pub is next to a house and every house is next to a pub, print
Impossible on a line. Otherwise output
$n$ space separated words.
Print pub or house for
each construction site. The first word indicates what to build
at construction site 1, the next at construction site 2, and so
on. If there are multiple valid solutions, you can output any
of them.